Asymptotic normality of a discrete procedure of stochastic approximation in a semi-Markov medium

We obtain sufficient conditions for the asymptotic normality of a jump procedure of stochastic approximation in a semi-Markov medium using a compensating operator of an extended Markov renewal process. The asymptotic representation of the compensating operator guarantees the construction of the generator of a limit diffusion process of the Ornstein-Uhlenbeck type. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1425–1433, October, 2006.     

Asymptotic normality of the continuous-time stochastic approximation algorithm

In this paper the continuous-time stochastic approximation algorithm seeking for the zero of a regression function is considered when the measurement error is a stochastic process generated by an Ito integral as the input of a linear system. The conditions are given to guarantee the asymptotic normality of the algorithm which is modified from the Robbins-Monro procedure proposed for the case where the measurement error is a process of independent increment.     

Continuous procedure of stochastic approximation in a semi-Markov medium

Using the Lyapunov function for an averaged system, we establish conditions for the convergence of the procedure of stochastic approximation

Asymptotic normality for discretely observed Markov jump processes with an absorbing state

For a continuous-time Markov process, occasionally, only discrete-time observations are available. For a simple sample of homogeneous Markov jump processes with an absorbing state, observed each on a stochastic grid of time points, we establish asymptotic normality of the maximum likelihood estimator and close the gap in Kremer and Weißbach (2013). By showing that the solution of the Kolmogorov backward equation system is continuous differentiable, we can apply results for M-estimators.     

Diffusion approximation of stochastic Markov models with persistent regression

Sequences of sums of identically distributed random variables forming a homogeneous Markov chain are approximated by a time-discrete autoregression process of Ornstein-Uhlenbeck type.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 7, pp. 928–935, July, 1995.     

Asymptotic behavior of truncated stochastic approximation procedures

We study asymptotic behavior of stochastic approximation procedures with three main characteristics: truncations with random moving bounds, a matrix-valued random step-size sequence, and a dynamically changing random regression function. In particular, we show that under quitemild conditions, stochastic approximation procedures are asymptotically linear in the statistical sense, that is, they can be represented as weighted sums of random variables. Therefore, a suitable formof the central limit theoremcan be applied to derive asymptotic distribution of the corresponding processes. The theory is illustrated by various examples and special cases.     

Asymptotic analysis of the behavior of a multichannel queueing system functioning in a Markov medium

The asymptotic behavior of some multidimensional characteristics of two Markov queueing systems, in which an incoming flow of units and their service time depend on a small parameter ɛ and the state of the Markov medium where these queueing systems function, is investigated. Bibliography: 6 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 91–98.     

Asymptotic confidence regions of stochastic approximation procedures in hilbert spaces

In the framework of stochastic approximation, in separable Hilbert spaces one can often establish weak convergence for a suitable normalized, sequence of random variables to a Gaussian distributed random varible. In connection with a sequence of empirical covariance operators and estimator of the unknown radius of a ball is described, for which the Gaussian limit distribution, takes a given value. Further a stopping rule is proposed leading to asymptotic confidence balls with a fixed radius.     

Asymptotic normality in a coupon collector's problem

Summary Let {a _s , s=1, 2, ..., N} be a set of reals and {p _s , s=1, 2, ..., N} be a set of probabilities, i.e. p _s0 and p ₁+p ₂+...+p _N =1. Let I ₁ I ₂,... be independent random variables, all with the distribution P(I=s)=p _s , s=1, 2, ..., N. Put U _v =l if I _v {I ₁, I ₂, ..., I _v –1} and U _v =0 otherwise, v=1, 2, .... The random variable Z _n = is called the bonus sum after ncoupons for a coupon collector in the situation {(p _s , a _s ), s=1, 2, ..., N}.Consider a sequence {(p _ks , a _ks ), s=l, 2, ..., N _k }, k=1, 2, ..., of collector situations, and let {Z _n ^(k) , n=1, 2, ...}, k=1, 2, ..., be the corresponding sequence of bonus sum variables. Let d be an arbitrary natural number and let , k=1, 2, ..., where 1 n _k ⁽¹⁾<n _k ⁽²⁾<< n _k ^(d) .We assume that N ^(k) t8 and that .It is shown that the random vector V ^(k) is, under general conditions, asymptotically (as kt8) normally distributed. An asymptotic expression for the covariance matrix of V ^(k) is derived.Research supported in part at Stanford University, Stanford, California under contract N0014-67-A-0112-0015.     

Strong representation of an adaptive stochastic approximation procedure

We consider a rather general one-dimensional stochastic approximation algorithm where the steplengths might be random. Without assuming a martingale property of the random noise we obtain a strong representation by weighted averages of the error terms. We are able to apply the representation to an adaptive process in the case where the random noise is a martingale difference sequence as well as in the case where the random noise is weakly dependent and some moment conditions are statisfied.     

Short-term memories with a stochastic perturbation

We investigate short-term memories in linear and weakly nonlinear coupled map lattices with a periodic external input. We use locally coupled maps to present numerical results about short-term memory formation adding a stochastic perturbation in the maps and in the external input.     

Asymptotic analysis of stochastic block replacement policies for multicomponent systems in a Markov environment

An asymptotic analysis of stochastic block replacement policies for a multicomponent system with fast Markov switches is provided. It is proved that this system can be approximated by a simpler system with exponential failures and averaged failure rates. The optimality of the long-run cost function is studied, as well.     

Asymptotic periodicity of a food-limited diffusive population model with time-delay

In this paper, a general reaction-diffusion food-limited population model with time-delay is proposed. Accordingly, the existence and uniqueness of the periodic solutions for the boundary value problem and the asymptotic periodicity of the initial-boundary value problem are considered. Finally, the effect of the time-delay on the asymptotic behavior of the solutions is given.     

Asymptotic normality and efficiency of a weighted correlogram

For a process X(t)=Σ _j=1 ^M g _j (t)ξ _j (), where g_j(t) are nonrandom given functions, is a stationary vector-valued Gaussian process, Eξ_k(t) = 0, and Eξ_k(0) Eξ_l(τ) = r _kl(τ), we construct an estimate for the functions r _kl(τ) on the basis of observations X(t), t ∈ [0, T]. We establish conditions for the asymptotic normality of as T → ∞. We consider the problem of the optimal choice of parameters of the estimate depending on observations. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 937–947, July, 1998.     

Asymptotic analysis of a perturbation problem

An asymptotic expansion is constructed for the solution of the initial-value problem

when t is restricted to the interval [0,T/ε], where T is any given number. Our analysis is mathematically rigorous; that is, we show that the difference between the true solution u(t,x;ε) and the Nth partial sum of the asymptotic series is bounded by ε^N+1 multiplied by a constant depending on T but not on x and t.     

Asymptotic normality in random graphs with given vertex degrees

We consider random graphs with a given degree sequence and show, under weak technical conditions, asymptotic normality of the number of components isomorphic to a given tree, first for the random multigraph given by the configuration model and then, by a conditioning argument, for the simple uniform random graph with the given degree sequence. Such conditioning is standard for convergence in probability, but much less straightforward for convergence in distribution as here. The proof uses the method of moments, and is based on a new estimate of mixed cumulants in a case of weakly dependent variables. The result on small components is applied to give a new proof of a recent result by Barbour and Röllin on asymptotic normality of the size of the giant component in the random multigraph; moreover, we extend this to the random simple graph.